Finding limit as n approaches infinity

I am not able to prove it by the hint shown. I perfectly understand the question and it is obvious that the antiderivative approaches 0 as n approaches infinity. But the hint about riemann integral is confusing me.

I am not able to prove it by the hint shown. I perfectly understand the question and it is obvious that the antiderivative approaches 0 as n approaches infinity. But the hint about riemann integral is confusing me.

The idea is that you want to prove that if , then uniformly on . Then you know that you can interchange the limits, i.e:

So all you need to prove is uniform convergence, and then the above equality will hold. (Hint: Use Dini's Theorem.)